A Gelfand-Tsetlin pattern is a type of combinatorial object that is useful in representation theory. We follow the definitions given in Molev, "Gelfand-Tsetlin bases for classical Lie algebras", 2018.
Let $\lambda$ be a partition with $n$ parts, written in nonincreasing order. A Gelfand-Tsetlin pattern of shape $\lambda$ is a triangular array of the following form:
$\begin{array}{ccccccccc} x_{n,1} & & x_{n,2} & & x_{n,3} & & \cdots && x_{n,n} \\ &x_{n-1,1} & & x_{n-1,2} & & \cdots & x_{n-1,n-1} & \\ && \ddots \\ && & x_{2,1} && x_{2,2} \\ &&&& x_{1,1}\end{array}$
Each entry $x_{i,j}$ is a nonnegative integer, the top row $x_{n,i}$ corresponds to $\lambda$, and the entries satisfy the inequalities $x_{k,i} \geq x_{k-1,i} \geq x_{k,i+1}$.
The Gelfand-Tsetlin patterns of shape $\lambda$ form a basis of the irreducible $sl_n$ character with highest weight $\lambda$, and there are explicit formulae for the actions of a basis of $sl_n$ on this basis.
The Gelfand-Tsetlin patterns correspond to the integer points of a polytope called the Gelfand-Tsetlin polytope. The LieAlgebraRepresentations package can create this polytope with the function gtPolytope.
A GTPattern is a hash table with keys recording the shape, entries, content, and weight of the pattern.
Currently only implemented for type A.